We cancheck(see section 21.8.5)that the number of states agrees with the number of product
states.
We have been expanding the states of definite total angular momentumjin terms of the product
states for several cases. The general expansion is called theClebsch-Gordan series:
ψjm=∑
m 1 m 2〈ℓ 1 m 1 ℓ 2 m 2 |jmℓ 1 ℓ 2 〉Yℓ 1 m 1 Yℓ 2 m 2or in terms of the ket vectors
|jmℓ 1 ℓ 2 〉=∑
m 1 m 2〈ℓ 1 m 1 ℓ 2 m 2 |jmℓ 1 ℓ 2 〉|ℓ 1 m 1 ℓ 2 m 2 〉The Clebsch-Gordan coefficients are tabulated. We have computedsome of them here by using the
lowering operator and some by making eigenstates ofJ^2.
21.6 Interchange Symmetry for States with Identical Particles
If we are combining the angular momentum from two identical particles, like two electrons in an
atom, we will be interested in the symmetry under interchange of the angular momentum state. Lets
use the combination of two spin^12 particles as an example. We know that we get total spin states
ofs= 1 ands= 0. Thes= 1 state is called atripletbecause there are three states with different
mvalues. Thes= 0 state is called asinglet. The triplet state is symmetric under interchange.
Thehighest total angular momentum state,s=s 1 +s 2 , will always besymmetric under
interchange.We can see this by looking at the highestmstate,m=s. To get the maximumm,
both spins to have the maximumzcomponent. So the product state has just one term and it is
symmetric under interchange, in this case,
χ 11 =χ(1)+χ(2)+.When we lower this state with the (symmetric) lowering operatorS−=S(1)−+S(2)−, the result
remains symmetric under interchange. To make thenext highest state,with two terms, we must
choose a state orthogonal to the symmetric state and this will always beantisymmetric.
In fact, for identical particles, thesymmetry of the angular momentum wave function will
alternate,beginning with a symmetric state for the maximum total angular momentum. For
example, if we add two spin 2 states together, the resulting statesare: 4S, 3A, 2S, 1Aand 0S. In
the language ofgroup theory,when we take thedirect product of two representations of
the the SU(2) groupwe get:
5 ⊗5 = 9S⊕ (^7) A⊕ (^5) S⊕ (^3) A⊕ (^1) S
where the numbers are the number of states in the multiplet.
- See Example 21.7.4:Two electrons in a P state.*
- See Example 21.7.5:The parity of the pion fromπd→nn.*